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A cubulation of a group is a metrically proper action by cubical automorphisms on a CAT(0) cube complex. It is cocompact when the action is also a cocompact group action.

The vertices of the dual cube complex are consistent choices of one halfspace of every wall that differ from a principal choice on only finitely many walls. Two vertices are joined when their choices differ on one wall, and higher cubes fill the resulting hypercubes. The result is a CAT(0) cube complex.

A group is cubulated when it admits a cubulation of a group, namely a metrically proper group action by cubical automorphisms on a CAT(0) cube complex. It is cocompactly cubulated when that action is also a cocompact group action. Thus a cocompact cubulation is a proper, cocompact cubical action.